3. The following table listed 25 subgroups of measurement data with group size 3

Question

3. The following table listed 25 subgroups of measurement data with group size 3 c1 49.431950.1432 49.2045 0.378950.2715 50.5952 48.6722 49.1436 49.1306 49.5332 50.4829 50.0007 49.486647.9221 50,4324 48.6924 50.1697 49.8926 50.8253 50.7303 50.8569 50.330649.8906 49.5988 49.9728 51.6716 50.7228 50.5934 50.7355 49.3386 49.554749.7753 49.7181 51.3211 48.048 51.7502 c1 50.317550.1224 50.4081 49.507749.7629 49.4091 49.3587 49.639 49.5229 49.500549.5897 49.7904 47.91149.5549 48.6716 49.885149.8281 49.8762 9.9994 50.6484 50,4223 49.58 49.4764 49.7899 51.10650.2333 50,4464 50.7538 49.9355 51.4371 49.7322 49.6442 50.2943 48.5373 51.9875 48.6112 48.6753 48.9048 49.3312 14 15 16 17 18 19 20 21 10 23 24 25 12 13 (a) The specification limits are 49 +- 3. Assume that the data is normally distributed! Estimate process capability ratio's (Cp, Cpk, Pp, Ppk). Is the process capable? (b) If the specification limits were 51 +/- 3. Re-calculate process capability ratio's (Cp, Cpk, Pp, Ppk). Is the process still capable? (c) Comment on the difference between Cp and Cpk ratios

Explanation / Answer

a. The specification limits are 49 +- 3. Assume that the data is normally distributed! Estimate process capability ratio's (Cp, Cpk, Pp, Ppk). Is the process capable?

We have, Mean µ = 49.8829

Standard deviation s (long term) = 0.5738

Standard deviation ? (short term) = 0.6037

Upper specification limit, USL = 49 + 3 = 52

Lower specification limit, LSL = 49 - 3 = 46

The process capability index,

Cp = (USL –LSL)/6? = (52 -46) /6*0.6037 = 1.66

Cpk= Min [(Mean – LSL)/3 *?, (USL – Mean)/3* ?]

= min [(49.8829 – 46)/ 3* 0.6037, (52 – 49.8829)/ 3 * 0.6037]

= min [(2.14, 1.17] = 1.17

Cpk = 1.17

Pp = (USL –LSL)/6s = (52 -46) /6*0.5738 = 1.7428

Ppk= Min [(Mean – LSL)/3 *s, (USL – Mean)/3* s]

= min [(49.8829 – 46)/ 3* 0.5738, (52 – 49.8829)/ 3 * 0.5738]

= min [(2.26, 1.23] = 1.23

Ppk = 1.23

Process is capable as process capability ratios are more than 1.

b. If the specification limits were 51 +/- 3. Re-calculate process capability ratio's (Cp, Cpk, Pp, Ppk). Is the process still capable?

We have, Mean µ = 49.8829

Standard deviation s (long term) = 0.5738

Standard deviation ? (short term) = 0.6037

Upper specification limit, USL = 51 + 3 = 54

Lower specification limit, LSL = 51 - 3 = 48

The process capability index,

Cp = (USL –LSL)/6? = (54 -48) /6*0.6037 = 1.66

Cpk= Min [(Mean – LSL)/3 *?, (USL – Mean)/3* ?]

= min [(49.8829 – 48)/ 3* 0.6037, (54 – 49.8829)/ 3 * 0.6037]

= min [(1.04, 2.27] = 1.04

Cpk = 1.04

Pp = (USL –LSL)/6s = (54 -48) /6*0.5738 = 1.7428

Ppk= Min [(Mean – LSL)/3 *s, (USL – Mean)/3* s]

= min [(49.8829 – 48)/ 3* 0.5738, (54 – 49.8829)/ 3 * 0.5738]

= min [(1.09, 2.39] = 1.09

Ppk = 1.09

Process is capable as process capability ratios are more than 1.

c. Comment on the difference between Cp and Cpk ratios

Difference between Cp and Cpk ration is an indicator that how far the average of the process is from the specifications.

# C1 C2 C3 Mean {= (C1 +C2 +C3)/3 = X Bar} R (max - min from C1,C2,C3) 1 50.3175 50.1224 50.4081 50.2827 0.2857 2 49.5077 49.7629 49.4091 49.5599 0.3538 3 49.3587 49.639 49.5229 49.5069 0.2803 4 49.5005 49.5897 49.7904 49.6269 0.2899 5 47.911 49.5549 48.6716 48.7125 1.6439 6 49.8851 49.8281 49.8762 49.8631 0.0570 7 49.9994 50.6484 50.4223 50.3567 0.6490 8 49.58 49.4764 49.7899 49.6154 0.3135 9 51.106 50.2333 50.4464 50.5952 0.8727 10 50.7538 49.9355 51.4371 50.7088 1.5016 11 49.7322 49.6442 50.2943 49.8902 0.6501 12 48.5373 51.9875 48.6112 49.7120 3.4502 13 48.6753 48.9048 49.3312 48.9704 0.6559 14 49.4319 50.1432 49.2045 49.5932 0.9387 15 50.3789 50.2715 50.5952 50.4152 0.3237 16 48.6722 49.1436 49.1306 48.9821 0.4714 17 49.5332 50.4829 50.0007 50.0056 0.9497 18 49.4866 47.9221 50.4324 49.2804 2.5103 19 48.6924 50.1697 49.8926 49.5849 1.4773 20 50.8253 50.7303 50.8569 50.8042 0.1266 21 50.3306 49.8906 49.5988 49.9400 0.7318 22 49.9728 51.6716 50.7228 50.7891 1.6988 23 50.5934 50.7355 49.3386 50.2225 1.3969 24 49.5547 49.7753 49.7181 49.6827 0.2206 25 51.3211 48.048 51.7502 50.3731 3.7022 Mean (X double bar; average of X bar) 49.8829 1.0221 R-bar St. Dev. (s) (Long term) 0.5738 d2 (for sample size 3) = 1.693 St. Dev. (? = R-bar/d2) (short term) 0.6037

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